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14h - 15h |
Salma Lahbabi (Paris-Dauphine & Université Hassan II de Casablanca & UM6P) |
Density functional theory for two dimensional homogeneous materials Abstract: We study Density Functional Theory models for 2D materials. We first show that a homogeneous material can be seen as a limit of crystals. Next, we derive reduced models in the orthogonal direction. We show how the different terms of the energy are modified and prove some properties of the ground state. In Kohn-Sham models, we prove that the Pauli principle is replaced by a penalization term in the energy. |
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15h15 - 16h15 |
Laurent Laflèche (Univ. Lyon 1) |
Quantum Optimal Transport and Sobolev Spaces Abstract: In the context of proving the semiclassical mean-field limit from the N-body Schrödinger equation to the Hartree-Fock and Vlasov equations, a crucial component is obtaining inequalities uniform in the Planck constant and the number of particles. These inequalities are the analogue of the estimates obtained in the corresponding kinetic models of classical statistical mechanics. Hence, in this presentation, I will introduce analogous tools and inequalities within the realm of quantum mechanics, such as operator versions of optimal transport and Sobolev spaces on the phase space, and the corresponding classical inequalities. We will in particular see that the quantum analogue of Sobolev inequalities yield uncertainty inequalities concerning the Wigner–Yanase skew information, and that the latter also plays a significant role in controlling a quantum Wasserstein "self-distance". |
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14h - 15h |
Søren Mikkelsen (Bath) |
Sharp semiclassical spectral asymptotics Abstract: In this talk we will discuss sharp semiclassical spectral asymptotics for differential operators, where the coefficients are not smooth. Usually, such results are proven using the theory of pseudo-differential operators. This theory is unfortunately not directly applicable, when we do not assume smoothness of the coefficients. How do we then obtain sharp semiclassical spectral asymptotics? The main focus of this talk will be to present the ideas in the proof of sharp semiclassical spectral asymptotics. |
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15h15 - 16h15 |
Ruoyu Wang (Univ. College London) |
Unbounded damped waves: backward uniqueness and polynomial stability Abstract: In this talk, we discuss the wave semigroup with an unbounded damping. In such a setting, there are explicit examples where the linear damped waves would go into finite-time extinction. We will then find an optimal condition explicitly on the unboundedness to guarantee that the finite-time extinction cannot happen. We will also develop powerful yet flexible control-theoretic tools to establish novel polynomial stability and energy decay results for a variety of damped wave-like systems, including the singular damped waves, the linearised gravity water waves and Euler-Bernoulli beams. |
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14h - 15h |
Jinyeop Lee (LMU, Munich) |
Derivation of the Vlasov equation from the fermionic many-body Schrödinger system using the Husimi measure Abstract: This seminar presents a derivation of the Vlasov equation from the fermionic many-body Schrödinger system, employing the Husimi measure as a liking tool. Our exploration begins with a heuristic understanding of the Vlasov equation's derivation. This is followed by a short review of the many-body Schrödinger equation. Then we will talk about the methodology of linking the solutions of the many-body Schrödinger equation with the Vlasov equation, namely the Wigner measure and the Husimi measure. Attendees will gain insights into the formalism of this approach and explore strategies for controlling residual terms appearing in the derivations. |
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15h15 - 16h15 |
Jacob Schach Møller (Aarhus) |
Weighted estimates for the Laplacian on the cubic metric lattice Abstract: We review how recent weighted resolvent estimates for the (discrete) Laplacian on Zd, together with a new embedding estimate, can be used to derive new weighted resolvent estimates for the Laplacian on the cubic metric lattice Ld = {x∈R d , for all but one j, x j ∈ Z}. In particular, some consequences for the spectrum of the (metric) Laplacian perturbed by suitable potentials will be discussed. The talk is based on joint work with E. Korotyaev and M. G. Rasmussen. |
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14h - 15h |
Léo Morin (U. de Copenhague) |
Quantum tunnelling between radial magnetic wells Abstract: This talk is devoted to the spectral analysis of 2D Schrödinger operators with inhomogeneous magnetic field. When the magnetic field has two symmetric minima, each well generates an eigenvalue of same order. Then, we expect tunnelling to happen: the two smallest eigenvalues are exponentially close to each other in the semiclassical limit. We prove this result in the case of radially symmetric wells. Based on the Helffer-Sjöstrand theory, we obtain an elegant and short proof. Even though non-magnetic tunnelling is already very well understood, magnetic fields come with specific issues. In particular, this is the first tunnelling result between purely magnetic wells, and the non-radial situation is still challenging. Finally, we observe some new purely magnetic effect on the interaction between the wells. |
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15h15 - 16h15 |
Olivier Bourget (UC Chile) |
On localization regimes for kicked quantum systems Abstract: We will discuss the spectral properties of some periodically kicked quantum systems defined on the lattice. We focus our analysis on the existence of (dynamical) localization regimes for a class of random perturbations. |
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14h - 15h |
Bérangère Delourme (Paris-Nord) |
Guided waves in honeycomb periodic structures Abstract: In this talk, we investigate the propagation of waves in a particular honeycomb structure made of thin tubes. Based on an asymptotic analysis, we prove that the dispersion surfaces associated with this system have conical points located at the vertices of the Brillouin zone: this is a well-known property for systems having hexagonal symmetry. Then, introducing so-called zig-zag perturbations in the structure generates guided waves propagating along the defect. More specifically, we show that the frequency of those guided modes may be independent of the quasi-periodicity parameter, leading to almost flat dispersion curves. We present numerical results to illustrate our results. |
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15h15 - 16h15 |
Asbjorn Bækgaard Lauritsen (IST Austria) |
Energies of dilute spin-polarized Fermi gasses Abstract: Recently the study of dilute quantum gases have received much interest, in particular regarding their ground state energies and pressures/free energies at positive temperature. I will present recent work on such problems concerning that of the ground state energy of a spin-polarized Fermi gas and the extension to the pressure at positive temperature. Compared to the free gas, the energy density/pressure of the interacting gas differs by a term of order a3 ρ8/3, with a the p-wave scattering length of the interaction. This talk is based on joint work with Robert Seiringer. |
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14h - 15h |
Vojkan Jaksic (Mc Gill) |
The phases of the number theoretic spin chain Abstract: The number theoretic spin chain was introduced in 1993 by Andreas Knauf in connection with Riemann zeta function, and has been intensely studied since. In this talk I will describe certain novel structural properties of this model and using them resolve the open problem regarding the structure of its phases at and above the critical temperature T=1/2. This talk is based on a joint work with T. Benoist, N. Cuneo, D. Jakobson, and C-A. Pillet. |
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15h15 - 16h15 |
Henrik Ueberschär (Sorbonne U.) |
Multifractality for periodic solutions of certain PDE Abstract: Many dynamical systems are in a state of transition between two regimes. Examples are firing patterns of neurons, disordered quantum systems or pseudo-integrable systems. A common feature which is often observed for critical states of such systems is a multifractal self-similarity in a certain scaling regime which cannot be captured by a single fractal exponent but only by a spectrum of fractal exponents. I will discuss a proof of multifractality of solutions for certain stationary Schrödinger equations with a singular potential on the square torus (joint with Jon Keating). Towards the end of the talk, I will allude to some new work on multifractal scaling and solutions to the Euler equations on cubic tori. |
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14h - 15h |
Frédéric Klopp (Sorbonne Univ.) |
The ground state of a system of interacting fermions in a random field: localization, entanglement entropy, ... Abstract: Transport in disordered solids is a phenomenon involving many actors. The motion of a single quantum particle in such a solid is described by a random Hamiltonian. Transport involves many interacting particles, usually, a small fraction of the particles present in the material. One striking phenomenon observed and proved in disordered materials is localization: disorder can prevent transport! While this is quite well understood at the level of a single particle, it is much less clear what happens in the case of many interacting particles. Physicists proposed a number of tools (exponential decay of finite particle density matrices, entanglement entropy, etc) to discriminate between transport and localization. Unfortunately, these quantities are very difficult to control mathematically for "real life" models. We will present a toy model where one can actually get a control on various of these quantities at least for the ground state of the system. The talk is based on the PhD theses of and joint works with N. Veniaminov and V. Ognov. |
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15h15 - 16h15 |
Christof Sparber (Univ. of Illinois at Chicago) |
Nonlinear bound states with prescribed angular momentum Abstract: We prove the existence of a class of orbitally stable bound state solutions to nonlinear Schrödinger equations with super-quadratic confinement in two and three spatial dimensions. These solutions are given by time-dependent rotations of a non-radially symmetric spatial profile which in itself is obtained via a doubly constrained energy minimization. One of the two constraints imposed is the total mass, while the other is given by the expectation value of the angular momentum around the z-axis. Our approach also allows for a new description of the set of minimizers subject to only a single mass constraint. This is joint work with I. Nenciu and X. Shen. |
Lundi 17 juin 2024
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14h - 15h |
Pablo Miranda (University of Santiago, Chile) |
Eigenvalue asymptotics for two dimensional magnetic Dirac operators Abstract: In this talk, we present results on the eigenvalue distribution for perturbed magnetic Dirac operators in two dimensions. We consider compactly supported perturbations and derive third-order asymptotic formulas that incorporate a geometric property of the perturbation's support. Notably, our approach allows us to consider some perturbations that do not necessarily have fixed sign, which is one the main novelties of our work. This is part of a joint work together with Vincent Bruneau. |
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15h15 - 16h15 |
Mi-Song Dupuy (Sorbonne Université) |
Linear response function for the computation of excited states in quantum chemistry Abstract: In quantum chemistry, a molecule at rest is described by its ground state, i.e., the eigenfunction corresponding to the smallest eigenvalue of an N-body Hamiltonian. The linear response function of interest in this talk is the one associated to one-body perturbations. This operator appears in various problems in quantum chemistry but is used in particular to compute energy differences via the poles of the linear response function. In practice, computing the linear response function for the N-body operator is too costly. One solution is to approximate it using the linear response function of a nonlinear evolution, which is the solution of a Dyson equation. The purpose of this talk is to introduce the linear response function in quantum chemistry, as well as the Dyson equation, and to study the poles of the solution of the Dyson equation. This is a joint work with Thiago Carvalho Corso (Univ. Stuttgart) and Gero Friesecke (TU Munich). |
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14h - 15h |
Pablo Miranda (University of Santiago, Chile) |
Eigenvalue asymptotics for two dimensional magnetic Dirac operators Abstract: In this talk, we present results on the eigenvalue distribution for perturbed magnetic Dirac operators in two dimensions. We consider compactly supported perturbations and derive third-order asymptotic formulas that incorporate a geometric property of the perturbation's support. Notably, our approach allows us to consider some perturbations that do not necessarily have fixed sign, which is one the main novelties of our work. This is part of a joint work together with Vincent Bruneau. |
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15h15 - 16h15 |
Pascal Millet (Sorbonne Paris-Nord) |
Spectral aspects of wave propagation on black hole spacetimes Abstract: The study of wave propagation on black hole spacetimes has been an active field of research in recent decades. This interest has been driven by the stability problem for black holes and by questions related to scattering theory. I will focus on the spectral approach, which relates the asymptotic properties of the solution to the properties of the resolvent operator. These properties are constrained by the geometric features of the black hole background and by the lower-order structure of the equation under consideration. I will illustrate this interplay, mainly in the case of the Teukolsky equation which governs the evolution of various massless fields around spinning black holes. In this context, resolvent analysis can be used to precisely determine the late-time asymptotics of initially localized solutions. |